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Hygro-thermo-mechanical analysis of spalling in concrete walls at hightemperatures as a moving boundary problem

Michal Beneš a, Radek Štefan b,⇑

a Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic b Department of Concrete and Masonry Structures, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic 

a r t i c l e i n f o

 Article history:Received 7 February 2014Received in revised form 7 January 2015Accepted 8 January 2015Available online 12 February 2015

Keywords:ConcreteHigh temperatureSpallingHygro-thermo-mechanical analysisMoving boundaryThermal stressPore pressureFinite element method

a b s t r a c t

A mathematical model allowing coupled hygro-thermo-mechanical analysis of spalling in concrete wallsat high temperatures by means of the moving boundary problem is presented. A simplified mechanicalapproach to account for effects of thermal stresses and pore pressure build-up on spalling is incorporatedinto the model. The numerical algorithm based on finite element discretization in space and the semi-implicit method for discretization in time is presented. The validity of the developed model is carefullyexamined by a comparison between the experimentally determined data stated in literature and theresults obtained from the numerical simulation.

 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Mathematical modelling seems to be an effective and powerfultool to simulate the heated concrete behaviour (see e.g.  [22]).Several models based on more or less general physical backgroundhave been developed to simulate transport processes in heatedconcrete (see [27] and references therein). All developed modelsbuild on a system of conservation of mass and energy, but differin the complexity of phase description of state of pore water aswell as chemical reactions and different physical mechanisms of coupled transport processes in a pore system. A descriptivephenomenological approach was used to explore hygro-thermal

processes in concrete exposed to temperatures exceeding 100

Cstarting with the Baz ant & Thonguthai model [5]. Here, the liquidwater and water vapour are treated as a single phase, moisture,and the evaporable water is assumed to be formed by the capillarywater only. The main advantages for usage of this approach is rel-ative simplicity and small number of parameters that can beobtained from experiments. However, in such a way it is impossi-ble to consider the effects of phase changes of water and theapplicability of the single-fluid-phase models for temperatures

above the critical point of water is disputed. These deficienciesled to the development of more detailed multi-phase description,see e.g. the works of   [21,13]   for specific examples. Coupledmulti-phase models reflect the multi-phase structure of concrete,interactions between phases, phase changes of fluids and solidsand non-linear couplings between thermal, hygral and mechanicalprocesses. However, such increase in complexity comes at theexpense of a large number of model parameters, which determina-tion can be hardly obtained directly from experiments. Moreover,multi-phase models are computationally expansive. Despite rapidprogress in computer technologies, complex models still exceedcapabilities of recently developed numerical algorithms and

computational hardware. Therefore, in this paper we will adopt apragmatic concept and consider the simplified model obtaineddirectly fromthe complex multi-phase description. Relative impor-tance of thermodynamic fluxes will be quantitatively evaluated atthe level of material point and these results will allow us to neglectless important transport phenomena without lost of capability torealistically predict behaviour of concrete at extremely hightemperatures.

High-temperature exposure of concrete can lead to the risk of concrete spalling and, consequently, to the damage of the entirestructure (see e.g.   [25] and references therein, for the examplessee [34,53,54]). It is generally accepted that the spalling process inrapidly heated concrete is caused by two main processes – increase

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.050

0017-9310/ 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +420 22435 4633; fax: +420 23333 5797.

E-mail address:  radek.stefan@fsv.cvut.cz (R. Štefan).

International Journal of Heat and Mass Transfer 85 (2015) 110–134

Contents lists available at  ScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e :   w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

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of pore pressure end development of thermal stresses – that mayact separately or, which is more likely, in a combined way (seee.g. [38, Section 4]).

In literature, we can find many criteria to asses the risk and, in

some cases, also the amount of concrete spalling (for a brief sum-mary of some of these criteria, see e.g.  [25,38]). In our previouswork   [7], we have employed a heuristic engineering approach,originally proposed by Dwaikat and Kodur   [15, Section 3.3], inwhich the spalling is supposed to occur if the effective pore pres-sure exceeds the temperature dependent tensile strength of con-crete. In the present paper, we extend this criterion in order totake into account not only the pore pressure (which seems to benot the dominant mechanism of spalling, as observed by recentexperimental investigations   [31,46]   and numerical simulations[48]) but also the thermal stress as a driving force of spalling.

The paper is organised as follows. In Section  2, we specifygeneral thermodynamical and mechanical assumptions onconcrete as a porous multi-phase medium to obtain a reasonably

simple but still realistic model to predict hygro-thermal behaviourof concrete at very high temperatures. This Section is concluded by

presentation of conservation of mass and thermal energy, carefullyderived in Appendix A by quantitative parameter analysis. In Sec-tions 3 and 4, constitutive relationships are discussed in detailsand suitable boundary conditions for description of transport pro-

cesses through the surfaces of the concrete wall are presented,respectively. Section 5 deals with the problem of moving boundaryfor spalling simulation. Our approach is based on the combinedeffect of pore pressure and thermal stresses in concrete under hightemperature exposure. In Section 6, the complex problem is formu-lated as a fully coupled system of highly nonlinear partial differen-tial equations (PDE’s) supplemented with appropriate boundaryand initial conditions. Based on the full FEM (in space) and semi-implicit (in time) discretization of the mentioned system of PDE’s,an in-house research MATLAB code has been developed for thesolution of the system of nonlinear algebraic equations. Thenumerical algorithm is presented in Section   7. Section  8 bringsthe complete list of material properties of moist concrete at hightemperatures used in the numerical model. Section  9 is the key

part of the paper. Here we present validation of the model bycomparison of the numerical results with experiments reported

Nomenclature

Symbol Descriptionac    convective heat transfer coefficient [W m2 K1]bc    convective mass transfer coefficient [m s1]tot    total strain [–]h   free thermal strain [–]

r   instantaneous stress-related strain [–]cr    creep strain [–]tr    transient strain [–]gs   volume fraction of the solid microstructure [–]gw   volume fraction of liquid phase [–]g g    volume fraction of gas phase [–]/   porosity [–]qs   density of the solid microstructure [kg m3]q g    gas phase density [kg m3]qa   dry air phase density [kg m3]q

v   vapour phase density [kg m3]

qv 1

  ambient vapour phase density [kg m3]qw   liquid phase density [kg m3]lw   liquid water dynamic viscosity [Pa s]l g    gas dynamic viscosity [Pa s]r   stress [Pa]rht    stress caused by hygro-thermal processes [Pa]rtm   stress caused by thermo-mechanical processes [Pa]rSB   Stefan–Boltzmann constant [W m2 K1]h   absolute temperature [K]h1

  ambient absolute temperature [K]kc    effective thermal conductivity of moist concrete

[W m1 K1]s   characteristic time of mass loss governing the asymp-

totic evolution of the dehydration process [s]c    mass of cement per m3 of concrete [kg m3]c w p   specific heat at constant pressure of liquid water

[J kg1 K1]c v 

 p   specific heat at constant pressure of water vapour[J kg1 K1]

c a p   specific heat at constant pressure of dry air [J kg1 K1]c s p   specific heat at constant pressure of solid phase [J kg1

K1]c  g  p   specific heat at constant pressure of gas mixture

[J kg1 K1]

e   emissivity of the interface [–] f c    uniaxial compressive strength of concrete [Pa] f t    uniaxial tensile strength of concrete [Pa]F    failure function [–]he   enthalpy of evaporation per unit mass [J kg1]hd   enthalpy of dehydration per unit mass [J kg1]H w specific enthalpy of liquid water [J kg1]H v  specific enthalpy of water vapour [J kg1]H s specific enthalpy of chemically bound water [J kg1] J w   liquid water mass flux [J kg1] J 

v   water vapour mass flux [kg m2 s1]

 J M    moisture flux [kg m2 s1]K    intrinsic permeability [m2]K rw   relative permeability of liquid water [–]K rg    relative permeability of gas [–]‘   thickness of a concrete wall [m]md   mass source term related to the dehydration process

[kg m3]md;eq   mass of water released at the equilibrium [kg m3]me   vapour mass source caused by the liquid water evapora-

tion [kg m3]M w   molar mass of liquid water [kg kmol1]M a   molar mass of dry air [kg kmol1]P    pore pressure due to water vapour (P  ¼ P v ) [Pa]P c    capillary pressure [Pa]P v    water vapour pressure [Pa]P a   dry air pressure [Pa]P  g    gas pressure [Pa]P w   pressure of liquid water [Pa]P s   water vapour saturation pressure [Pa]qc    heat flux vector [W m2]R   gas constant [J kmol1 K1]RH    relative humidity (RH ¼ P v =P s) [–]S w   degree of saturation with liquid water [–]S B   degree of saturation with adsorbed water [–]S ssp   solid saturation point [–]v  g    velocity of gaseous phase [m s1]v w   velocity of liquid phase [m s1]v a   velocity of dry air [m s1]v v    velocity of vapour [m s1]

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in literature by means of three examples. First two examples, per-formed by Kalifa et al.  [36] and, more recently, [44], examine thecomparison of the measurements of pore pressure and tempera-ture distributions with the predicted numerical simulations basedon the present model under specific conditions excluding spallingphenomena. Finally, we apply the present model to investigate thesurface spalling of high strength concrete prismatic specimenunder unidirectional heating by the ISO 834 fire curve and comparethe numerical results with experimental observations reported byMindeguia [44] and Mindeguia et al.  [45,46]. The summary of theoutcomes achieved in this paper as well as the general conclusionsand recommendations for future research appear in Section 10.

2. Basic conservation equations

 2.1. General assumptions

A number of assumptions have been adopted to develop thecoupled hygro-thermo-mechanical model for concrete spallingdue to high temperatures exposure. Some of them of particularinterest are as follows:

 concrete is assumed as a multi-phase system consisting of different phases and components: solid skeleton (composed of various chemical compounds and chemically bound water),liquid phase (combined capillary and adsorbed water), gasphase (a mixture of dry air and water vapour) (see e.g.  [21]);

  the diffusive mass flux of water vapour is neglected and theadsorbed water diffusion is assumed to be expressed by theliquid water relative permeability term,   K rw, instead of aseparate term (see [10,13]);

 the effects of variations of pressure of dry air is neglected. Themass of dry air in concrete is considered to be much smallerthan the mass of liquid water and vapour and, consequently,the vapour pressure plays the crucial role (when compared todry air) in spalling phenomena (see [15]);

 above the critical point of water, only the vapour contributionto the mass transport is taken into account (cf.  [23]);

 the spalling of concrete in a heated wall is assumed to be causedby a combination of the hygro-thermal stress due to the porepressure build-up and the thermo-mechanical stress resultingfrom the restrained thermal dilation (see e.g. [38]);

 the wall is considered as fully mechanically restrained in theplane perpendicular to the wall thickness (see [48,58]);

 the stresses resulting from the external mechanical load as wellas the self-weight of the wall are not accounted for sincetheir effect on the potential spalling may be considered to benegligible compared with the effect of mechanical restraint(see [48,58]);

 the hygro-thermal and the mechanical problems are coupled in

the analysis.

 2.2. Conservation laws

The mathematical model of moisture and heat transfer in con-crete consists of the balance equations governing the conservationof mass (moisture) and thermal energy (cf. Eqs. (A.9) and (A.23)).

The mass balance equation of moisture (liquid water and vapour):

@ 

@ t ðgwqw þ g

v q

v Þ þ   @ 

@  x  gwqwv w þ g

v q

v v v 

¼ @ md

@ t   ;   ð1Þ

energy conservation equation for moist concrete as the multi-phasesystem:

ðqc  pÞ@ h

@ t  ¼ @ qc 

@  x  ðqc  pv Þ

@ h

@  x @ me

@ t   he @ md

@ t   hd;   ð2Þ

where

ðqc  pÞ ¼ c w p  qw gw þ c  g  pq g  g g  þ c s p qs gs;   ð3Þ

ðqc  pv Þ ¼ c w p qwgwv w þ c  g  pq g g g v  g    ð4Þ

and  c  g  pq g  ¼ c v 

 pqv  þ c a pqa. Governing Eqs.  (1) and (2)   are carefully

derived in   Appendix A   and the meanings of all symbols areexplained in a well arranged way in Nomenclature. It should be

underlined that material coefficients of concrete and fluids thatappear in governing Eqs. (1) and (2) depend in non-linear mannerupon the primary unknowns – temperature and pore pressure,which completely describe the state of concrete under thermalloading.

3. Constitutive relationships

Balance Eqs. (1) and (2) are supplemented by an appropriate setof constitutive equations.

Moisture flux.  As the constitutive equations for fluxes of fluidphases (liquid water and vapour) the Darcy’s law  is applied

gwqwv w

 ¼ qw

KK rw

lw

@ P w

@  x

  ;   P w ¼

P 

P c ;

  ð5

Þg

v q

v v v  ¼ q

v 

KK rg 

l g 

@ P 

@  x;   ð6Þ

where P w [Pa] is the pressure of liquid water, P c  [Pa] is the capillarypressure and P  [Pa] represents the pore pressure due to the watervapour. Further,   K  ½m2   represents the intrinsic permeability,K rw½ and K rg ½ are the relative permeability of liquid water andrelative permeability of gas,  lw   [Pa s] and  l g  [Pa s] represent the

liquid water and gas dynamic viscosity.The equilibrium state of the capillary water with the water

vapour is expressed in the form of the  Kelvin equation

P c  ¼ qw

hR

M wln

  P 

P s ;   ð7Þ

where   P c   denotes the capillary pressure and the water vapoursaturation pressure  P s  [Pa] can be calculated from the followingformula as a function of temperature  h  [K]

P sðhÞ ¼ exp 23; 5771   4042; 9

h 37; 58

:   ð8Þ

Water vapour is considered to behave as perfect gas, therefore,Clapeyron equation [21]

qv  ¼ PM w

hR  ð9Þ

is assumed as the state equation, where  R ½ J mol1 K1   is the gas

constant   ð8:3145 J mol1 K1Þ  and M w  ½kg mol1  represents molar

mass of water vapour.Heat flux. For the heat flux induced by conduction the Fourier’s

law is applied in the form

qc  ¼ kc 

@ h

@  x;   ð10Þ

where   kc  ½W m1 K1   represents the temperature and saturationdependent effective thermal conductivity of moist concrete.

Evaporation.  In order to determine the amount of heat due toevaporation or, reversely, condensation processes, the watervapour conservation equation

@ ðgv q

v Þ

@ t   þ @   g

v q

v v v 

@  x

  ¼ @ me

@ t   ð11Þ

needs to be incorporated into the energy conservation Eq. (2) whichwill be handled in Section 6.2.

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Dehydration.   Following  [11,19], the evolution of mass sourceterm md [kg m3] related to the dehydration process is consideredthrough the following evolution law

@ md

@ t   ¼ 1

sðmd md;eqðhÞÞ;   ð12Þ

where md;eq   [kg m3] is the mass of water released at the equilib-rium according to temperature  h  and s [s] is the characteristic timeof mass loss governing the asymptotic evolution of the dehydrationprocess.

4. Boundary conditions

To describe coupled transport processes through the surfaces of the wall, one should prescribe the appropriate boundary condi-tions across the boundary. Homogeneous Neumann conditions

heqv g

v v v   kc 

@ h

@  x¼ 0;   ð13Þ

qwgwv w þ qv g

v v v  ¼ 0   ð14Þ

are usually applied on the insulated surface of the wall. Boundaryconditions of the form

heqv g

v v v   kc 

@ h

@  x

n x ¼ ac ðh h1Þ þ erSBðh4 h4

1Þ;   ð15Þ

ðqwgwv w þ qv g

v v v Þn x ¼ bc ðqv 

  qv 1Þ ð16Þ

(n x  ¼ 1) are of importance on the exposed side of the wall, wherethe terms on the right hand side of  (15) represent the heat energydissipated by convection and radiation to the surrounding mediumand the term on the ride hand side of   (16)   represents a watervapour dissipated into the surrounding medium.

5. Moving boundary for spalling simulation in concrete walls

5.1. Spalling criterion

Let us assume a concrete wall of a thickness ‘ exposed to fire onboundary x ¼ ‘. The spalling at position x 2 ð0; ‘Þ and time t  occursif (cf. [48, p. 613]; [51, Section 2.3])

F ð f c ðhÞ; f t ðhÞ;rht ðP ; hÞ;rtmðhÞÞ > 1;   ð17Þwhere   F   is a dimensionless failure parameter (failure function), f c   and   f t    are the temperature dependent uniaxial compressiveand tensile strengths of concrete, respectively, rht   and rtm  are theactual stresses in concrete caused by hygro-thermal and thermo-mechanical processes, respectively, and   P   and   h   are the porepressure and temperature in concrete at position   x   and time   t ,respectively.

Note that the strengths of concrete (both compressive andtensile) are assumed to be positive values while the strains and

stresses are taken as positive in tension and negative incompression.

5.2. Hygro-thermal stress

Hygro-thermal stress in heated concrete is a tensile stresscaused by the pore pressure build-up. There are several approachesto determine the hygro-thermal stress. These approaches may dif-fer both in the definition of the pore pressure and also in the man-ner in which the pore pressure is converted into the hygro-thermalstress. The pore pressure may be assumed to be equal to thevapour pressure (e.g. [15]), to the gas pressure, or can be calculatedas the Bishop’s stress  [12,21,48,51]. The hygro-thermal stress (i.e.the effective pore pressure) can be determined from the pore pres-

sure by the hollow spherical model   [30], or multiplying respec-tively by a Biot’s coefficient [48,51] or by a concrete porosity [15].

Here, we adopt the approach proposed by Dwaikat and Kodur[15], in which

rht ðP ; hÞ ¼ P /ðhÞ;   ð18Þwhere P  is the pore pressure due to water vapour and  / is the tem-perature dependent concrete porosity.

5.3. Thermo-mechanical stress

The thermo-mechanical stress in a heated concrete wallgenerally depends on the external mechanical load applied onthe wall, on its geometry (wall thickness, load eccentricity),material properties (both hygro-thermal and mechanical), andboundary conditions (heating, mechanical restraint).

In our approach, we follow a conservative assumption, alsoadopted by e.g. [48,58], in which the wall is supposed to be fullymechanically restrained in the plane perpendicular to the wallthickness. On the other hand, the stresses resulting from theexternal mechanical load as well as the self-weight of the wallare not accounted for since their effect on the potential spallingmay be considered to be negligible when compared with the effect

of mechanical restraint (cf. [48,58]).In order to assess the thermal stress arising from the mechani-cal restraint, we have to focus on the stress–strain conditions in theheated wall. It is widely accepted that the total strain in concretesubjected to high temperatures may be decomposed into severalparts which differ in their physical meaning. Hence, we can write[2, Eq. (1)]; [3, Eq. (4.2)]; [41, Eq. (1)]

tot  ¼ hðhÞ þ rðr; hÞ þ cr ðr; h; t Þ þ tr ðr; hÞ;   ð19Þwhere tot  is the total strain, h  is the free thermal strain, r  is theinstantaneous stress-related strain (which can be divided into theelastic part and the plastic part,  e   and  p, respectively (see e.g.[8])), cr  is the creep strain, tr  is the transient strain, r is the stress,and t  is the time.

As stated in e.g. [28], the creep strain, cr , is usually neglected.Moreover, the stress-dependent strains can be assumed together(with or without r  or cr ) as the mechanical strain (see  [28]) oras the so called load induced thermal strain – LITS (see e.g.[2,39,56,58]).

Here, we adopt a constitutive law proposed by Eurocode 2 [17],in which the stress is expressed in terms of the total mechanicalstrain, m, that includes the transient strain implicitly, while thecreep strain is omitted [2,9,28], and hence, we can write

tot  ¼ hðhÞ þ mðr; hÞ:   ð20ÞSince the wall is supposed to be fully mechanically restrained,

the total strain is equal to zero and the mechanical strain can beexpressed as

m ¼ hðhÞ:   ð21ÞAs mentioned above, the constitutive law given by Eurocode 2

[17] has the form

r ¼ Lðm; hÞ:   ð22ÞIn our case (see Eq. (21)), the stress can be expressed directly as

a function of temperature (the formulas provided by Eurocode 2[17]  for hðhÞ   and   Lðm; hÞ  are stated in Section  8). It should benoted that the stress in concrete determined by (22)  belongs tothe uniaxial conditions. For the plane stress conditions, we get

rtmðhÞ ¼   1

1 mðhÞ  rðhÞ;   ð23Þ

where m  is the Poisson’s ratio (see Section 8), and r  is the uniaxialstress given by (22) (with m  determined by (21)).

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5.4. Resulting stress conditions

Let us assume that the wall thickness,  ‘, which is much smallerthan the other dimensions of the wall, is parallel to the x-axis, andthe wall surfaces (at positions of  x ¼ 0 and x ¼ ‘) are parallel to theplane y- z  (cf. [48]). As mentioned above and as stated in [48], wecan suppose that the hygro-thermal stress acts as a tensile stressperpendicular to the wall surface (i.e. in the x-direction) and thethermo-mechanical stress acts as a compressive plane stress inthe planes parallel to the wall surface (i.e. in the directions of  yand   z ), see   Fig. 1. Hence, in terms of the principal stresses,r1;r2;r3, and the normal stresses,  r x;r y;r z , we may write [48]

r1 ¼ r x ¼ rht ðP ; hÞ;   ð24Þ

r2 ¼ r3 ¼ r y ¼ r z  ¼ rtmðhÞ:   ð25Þ

5.5. Failure function

As a failure function, we adopt the Menétrey–Willam triaxialfailure criterion defined as [43, Eq. (6)]

F  ¼  ffiffiffiffiffiffiffi1:5p    q

 f c 

2

þ   3 f 

2c   f 

2t 

 f c  f t 

eF 

eF  þ 1

!  q ffiffiffi

6p 

  f c 

r ð#; eF Þ þ   n ffiffiffi3

p   f c 

!;   ð26Þ

where eF   is the dimensionless eccentricity (0:5 < eF  6 1:0), whichinfluences the shape of the failure function (see e.g.   [43, Fig. 3])and can be expressed as a function of  f c ; f t   and f b, with f b   beingthe biaxial compressive strength of concrete (see   [33, Eq.(21.28)]). Further, function r ð#; eF Þ is given by Menétrey and Willam[43, Eq. (1)] in the form

r ð#; eF Þ ¼   4ð1 e2F Þ cos

2

# þ ð2eF   1Þ2

2ð1 e2F Þ cos # þ ð2eF   1Þ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ð1 e2

F Þ cos2 # þ 5e2F   4eF 

q ð27Þ

and   n;q   and  #   are the Haigh–Westergaard coordinates (see e.g.[33,43]).

The Menétrey–Willam failure function for the plane stress con-ditions (i.e. for r3  ¼ 0) is shown in Fig. 2.

In our case, in which r1  ¼ rht  and r2 ¼ r3  ¼ rtm, we can write

n ¼   1 ffiffiffi3

p   ðrht  þ 2rtmÞ;   ð28Þ

q¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3 ðrht 

rtm

Þ2r    ;

  ð29

Þcosð3#Þ ¼   rht  rtm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðrht  rtmÞ2q    :   ð30Þ

Since rtm  6 0 < rht , and 0:5 < eF  6 1:0, it is obvious that  # ¼ 0,and r  ¼ 1=eF . Substituting this in (26) and assuming the strengthsof concrete as temperature dependent, we get the resulting failurefunction used in our model (cf. [48, p. 613])

F  ¼   rht ðP ; hÞ rtmðhÞ f c ðhÞ

2

þ f c ðhÞ2  f t ðhÞ2

 f c ðhÞ2 f t ðhÞ

rht ðP ; hÞ þ 2eF   1

eF  þ 1 rtmðhÞ

;   ð31Þ

which is illustrated in Fig. 3.Fig. 1.   Stress conditions in the wall.

Fig. 2.  Menétrey–Willam failure function [43].

Fig. 3.  Failure function used in our model.

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5.6. Evolution law for moving boundary due to spalling 

In our regularized approach, the instantaneous spalling of concrete is approximated by rapid continuous process in such away that sheet of concrete is continuously removed from the walland the receding outer surface forms the moving boundary. Ingeneral, in one dimension, the external loading to the concrete wallaccording to the boundary conditions  (15) and (16) is prescribedon the unknown ablating boundary   x ¼ ‘ðt Þ. For the spalling of the wall originally occupying the space 0 6 x 6 ‘0, we proposethe governing equation of the form

d‘

dt  ¼ ‘

c ½maxðF Þ 1þ; ‘ð0Þ ¼ ‘0;   ð32Þ

where the position of the moving boundary ‘ has to be determinedas a part of the solution. Further, xþ ¼ maxð0; xÞ; c [s] represents thecharacteristic time of spalling process and maxðF Þ  is the maximalvalue of failure function  (31)   achieved within the wall   ð0; ‘Þ   atactual time t .

6. Formulation of the problem

6.1. Modification of mass balance equation

Let us start with modification of the Eq.   (1).   Denote bym ¼ gwqw þ g

v q

v   the total mass of moisture (including liquid

water and vapour). Incorporating the constitutive relations  (5)and (6) into the mass balance Eq.  (1) yields

@ m

@ t  ¼   @ 

@  x  qw

KK rw

lw

þ qv 

KK rg 

l g 

!@ P 

@  x qw

KK rw

lw

@ P c 

@  x

" #þ @ md

@ t   :   ð33Þ

Here P c  ¼ P c ðP ; hÞ via Eq. (A.14). After additional modification, theEq. (33) can be written in a general form

@ m@ t   @ md

@ t   ¼   @ 

@  x   K mP  @ P @  xþ K mh @ h

@  x ;   ð34Þ

where

K mP  ¼ qw

KK rw

lw

1 @ P c 

@ P 

þ q

v 

KK rg 

l g 

;   ð35Þ

K mh ¼ qw

KK rw

lw

@ P c 

@ h :   ð36Þ

6.2. Modification of energy conservation equation

Incorporating water vapour conservation equation into theterm corresponding to the latent heat of evaporation leads to thefollowing modified energy balance equation

ðqc  pÞ @ h

@ t  þ he

@   gv q

v 

@ t 

  þ hd

@ md

@ t 

¼   @ 

@  x  kc 

@ h

@  x

he

@   gv q

v v v 

@  x

  c w p qwgwv w þ c  g  pq g g g v  g 

@ h

@  x:   ð37Þ

Simple calculation yields

he

@   gv q

v v v 

@  x

  ¼ @   hegv q

v v v 

@  x

    gv q

v v v 

@ he

@  x

¼  @ 

@  x  heqv 

KK rg 

l g 

@ P 

@  x

!þ q

v 

KK rg 

l g 

@ he

@ h

@ P 

@  x

@ h

@  x:   ð38Þ

Incorporating the Eq. (38) into (37) reads

M hP 

@ P 

@ t  þ M hh

@ h

@ t  þ hd

@ md

@ t   ¼   @ 

@  x  K hP 

@ P 

@  xþ K hh

@ h

@  x

þ   C hP 

@ P 

@  xþ C hh

@ h

@  x

@ h

@  x;   ð39Þ

where

M hP  ¼ he

@   gv q

v  @ P    ;   ð40Þ

M hh ¼ ðqc  pÞ þ he

@   gv q

v 

@ h

  ;   ð41Þ

K hP  ¼ heqv 

KK rg 

l g 

;   ð42Þ

K hh ¼ kc ;   ð43Þ

C hP  ¼ c w p qw

KK rw

lw

1 @ P c 

@ P 

þ   c v 

 p @ he

@ h

q

v  þ c a pqa

KK rg 

l g 

;   ð44Þ

C hh ¼ c w p qw

KK rw

lw

@ P c 

@ h  :   ð45Þ

6.3. Resulting model

The full mathematical model consists of the balance equationsfor moisture and energy, state equation of pore water (moisture),governingequation for dehydration, evolution equation for movingboundary due to spalling, the set of appropriate boundary and ini-tial conditions specifying the fields of pore pressure, temperature,mass of dehydrated water and initial thickness of the wall.

Moisture conservation equation:

@ m

@ t   @ md

@ t   ¼   @ 

@  x  K mP 

@ P 

@  xþ K mh

@ h

@  x

;   ð46Þ

Energy conservation equation:

M hP @ P @ t 

 þ M hh@ h@ t 

 þ hd@ md

@ t   ¼   @ 

@  x  K hP 

@ P @  x

þ K hh@ h@  x

þ   C hP 

@ P 

@  xþ C hh

@ h

@  x

@ h

@  x;   ð47Þ

State equation of moisture:

m ¼ gwqw þ gv q

v ;   ð48Þ

Governing equation of dehydration:

@ md

@ t   ¼ 1

sðmd md;eqðhÞÞ;   ð49Þ

Evolution law for moving boundary due to spalling:

d‘

dt  ¼ ‘

c ½maxðF Þ 1þ;   ð50Þ

Boundary conditions (at x ¼ 0 and x ¼ ‘):

  K mP 

@ P 

@  xþ K mh

@ h

@  x

n x ¼ bc ðqv 

  qv 1Þ;   ð51Þ

  K hP 

@ P 

@  xþ K hh

@ h

@  x

n x ¼ ac ðh h1Þ þ erSBðh4 h4

1Þ ð52Þ

and initial conditions (at t ¼ 0):

P  ¼ P 0;   ð53Þh ¼ h0;   ð54Þ‘

¼‘0;

  ð55

Þmd ¼ 0:   ð56Þ

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The unknowns in the model are the moisture content  m, porepressure P , temperature  h , mass of dehydrated water md  and theactual thickness of the wall   ‘. Transport coefficientsK mP ; K mh; M hP ; M hh; K hP ; K hh; C hP ; C hh, defined by  (35) and (36)   and(40)–(45), depend in non-linear manner upon temperature  h andpressure   P . Material non-linearities are described in detail inSection 8.

7. FEM formulation and solution strategy 

Let 0 ¼ t 0  <  t 1 < < t N  ¼ T  be an equidistant partitioning of time interval   ½0; T   with step   Dt . Set a fixed integer  n  such that

0 6 n < N . In what follows we abbreviate   f ð x; t nÞ   by   f n for anyfunction   f . The time discretization of the continuous model isaccomplished through a semi-implicit difference scheme

mnþ1 mn

Dt   ¼   @ 

@  x  K nmP 

@ P nþ1

@  x  þ K nmh

@ hnþ1

@  x

!þ mnþ1

d  mn

d

Dt   ;   ð57Þ

M nhP 

P nþ1 P n

Dt   þ M nhh

hnþ1 h

n

Dt   ¼   @ 

@  x  K nhP 

@ P nþ1

@  x  þ K nhh

@ hnþ1

@  x

!

þ   C nhP 

@ P n

@  x þ C nhh

@ hn

@  x

@ hn

@  x

hnd

mnþ1d   mn

d

Dt   ;   ð58Þ

mnþ1 ¼ gnþ1w   qwðhnþ1Þ þ gnþ1

v   q

v ðhnþ1; P nþ1Þ;   ð59Þ

mnþ1d

  mnd

Dt   ¼ 1

sðmn

d md;eqðhnÞÞ;   ð60Þ‘nþ1 ‘n

Dt   ¼ ‘nþ1

c  ½maxðF nþ1Þ 1þ;   ð61Þ

n x   K nmP 

@ P nþ1

@  x  þ K nmh

@ hnþ1

@  x

! x¼0;  x¼‘nþ1

¼ bc    qv ðP nþ1; hnþ1Þ q

v ðP nþ1

1   ; hnþ11   Þ

;   ð62Þ

n x   K nhP 

@ P nþ1

@  x  þ K nhh

@ hnþ1

@  x

! x¼0;  x¼‘nþ1

¼ ac ðhnþ1 hnþ11   Þ þ erSB   ðhnþ1Þ4 ðhnþ1

1   Þ4

;   ð63Þ

where   gnþ1w   ¼ gwðhnþ1; P nþ1Þ   and   gnþ1

v   ¼ g

v ðhnþ1; P nþ1Þ. Further,

n x  ¼ þ1 for x ¼ ‘nþ1 and n x  ¼ 1 for x ¼ 0.Applying the Galerkin procedure to the mass and energy

conservation equations leads to the system of non-linear algebraicequations

1

Dt M

n

 x

n

þ1

 x

n þK

n

 x

n

þ1

þ f 

n

þ1

ð xn

þ1

Þ ¼0

;   ð64Þwhere   xnþ1 ¼   mnþ1; hnþ1; P nþ1

T ,   mnþ1ð xÞ ¼ Nð xÞmnþ1,   hnþ1ð xÞ ¼

Nð xÞhnþ1 and P nþ1ð xÞ ¼ Nð xÞ P nþ1, stores the unknown nodal valuesof moisture, temperature and pore pressure at time  t nþ1, respec-tively. The constant matrices in (64) exhibit a block structure

Mn ¼

Mnmm   0 0

0 Mnhh   M

nhP 

0 0 0

264

375;   K

n ¼0 K

nmh   K

nmP 

0 Knhh   K

nhP 

0 0 0

264

375   ð65Þ

and the non-linear term reads as

 f nþ1

ð xnþ1

Þ ¼

 f nþ1m   ð xnþ1Þ

 f nþ1h

  ð xnþ1

Þ f nþ1P    ð xnþ1Þ

0B@

1CA:

  ð66

Þ

The non-linear system (64) is solved iteratively using the Newton’smethod (see [52]).

The individual matrices M  can be expressed as

Mnmm ¼

Z   ‘nþ1

0

Nð xÞTNð xÞ d x;   ð67Þ

M

nhh ¼ Z   ‘nþ1

0Nð xÞ

T

ðqc  pÞ þ he

@   gv q

v  @ h

Nð xÞ d x;   ð68Þ

MnhP  ¼

Z   ‘nþ1

0

Nð xÞThe

@   gv q

v 

@ P 

  Nð xÞ

d x;   ð69Þ

whereas the blocks K  attain the form

Knmh ¼

Z   ‘nþ1

0

@ 

@  xNð xÞ

T

qw

KK rw

lw

@ P c 

@ h

@ 

@  xNð xÞ

d x;   ð70Þ

KnmP ¼

Z   ‘nþ1

0

@ 

@  xNð xÞ

T

qw

KK rw

lw

1@ P c 

@ P 

þq

v 

KK rg 

l g 

! @ 

@  xNð xÞ

" #d x;

ð71

Þ

Knhh ¼

Z   ‘nþ1

0

@ 

@  xNð xÞ

T

kc 

@ 

@  xNð xÞ

d x;   ð72Þ

KnhP  ¼

Z   ‘nþ1

0

@ 

@  xNð xÞ

T

heqv 

KK rg 

l g 

@ 

@  xNð xÞ

" #d x:   ð73Þ

The non-linear terms f  are provided by

 f nþ1m   ð xnþ1Þ ¼

Z   ‘nþ1

0

mnþ1d

  mnd

Dt   Nð xÞT d x

þ   bc    qv ðP nþ1; hnþ1Þ q

v ðP nþ1

1   ; hnþ11   Þ

Nð xÞT

h i x¼‘nþ1

 x¼0;

ð74Þ

 f nþ1h   ð xnþ1Þ¼

Z   ‘nþ1

0

hnd

mnþ1d

  mnd

Dt   Nð xÞT d x

Z   ‘nþ1

0

C nhP 

@ P n

@  x þC nhh

@ hn

@  x

@ hn

@  x Nð xÞT d x

þ   ac ðhnþ1 hnþ11   ÞþerSB   ðhnþ1Þ4 ðhnþ1

1   Þ4

Nð xÞT

h i x¼‘nþ1

 x¼0;

ð75Þ

 f nþ1P    ð xnþ1Þ ¼ mnþ1 gnþ1

w   ðhnþ1; P nþ1Þqwðhnþ1Þ gnþ1

v   ðhnþ1; P nþ1Þq

v ðhnþ1; P nþ1Þ:   ð76Þ

Numerical algorithm.   The described numerical algorithmbecomes:

Step 1. Set T ; Dt ; N ¼ T =Dt Step 2. Set initial values P 0; h0; ‘0; mdð0Þ ¼ 0Step 3.

For n ¼ 0; . . . ; N  1t n ¼ nDt 

update md;eqðhnÞ; F nþ1

solve (61) to get ‘nþ1

solve (60) to get mnþ1d

update P nþ11   ; hnþ1

1

update Mn;Kn; f nþ1

solve (64) by Newton’s iteration procedure to get  xnþ1

end n

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8. Material data for concrete at high temperatures

In our approach, concrete is assumed to be a homogeneousmulti-phase system. Hence, most of its material properties (suchas free thermal strain, thermal conductivity, etc.) are treated asso called effective or smeared characteristics. It means that forthe solid skeleton, we do not distinguish between the aggregates

and the cement paste. Such macroscopic approach has been widelyaccepted by the scientific community for investigation of hygro-thermo-mechanical and spalling behaviour of heated concrete(see e.g. the works by Dwaikat and Kodur [15], Gawin et al.  [25],Witek et al. [57]). It is clear that by turning to a mesoscale level,one can obtain more realistic results, especially when investigatingthe spalling phenomenon (see e.g. the recent works by Le [40] andZhao et al.   [59]). However, such approach is out of scope of thepresent paper.

Based on the literature review, the material properties of con-crete and its components subjected to high temperatures areassumed as follows.

The free thermal strain of concrete, h [–], is defined in the tem-perature range of 293:15 K to 1473:15 K as [17, Section 3.3.1].

For siliceous aggregates concrete:

hðhÞ¼   1:8104 þ9106hþ2:31011

h3 for   h6973:15K;

14103 for   h> 973:15K

(

ð77Þ

for calcareous aggregates concrete:

hðhÞ¼   1:2104 þ6106hþ1:41011

h3 for   h61078:15K;

12103 for   h> 1078:15K:

(

ð78ÞThe constitutive law for concrete in compression,  r ¼ Lðm; hÞ,

is adopted from Eurocode 2 in the form [17, Section 3.2.2.1]

rðm; hÞ ¼   3m f c ðhÞ

c 1ðhÞ   2þ   mc 1 ðhÞ

3   for   cu1ðhÞ < m  6 0;

0 for   m  6 cu1ðhÞ;

8>><>>: ð79Þ

where   m   is the total mechanical strain,   f c   is the compressivestrength of concrete, c 1  is the strain corresponding to f c , and cu1

is the ultimate strain, see Fig. 4 and Fig. 5.The temperature dependent parameters of the constitutive law

(79), f c ðhÞ; c 1ðhÞ, and cu1ðhÞ, for the normal strength concrete withthe siliceous and calcareous aggregates, NSC-S and NSC-C, respec-tively, as well as for three classes of high strength concrete (HSC-1,HSC-2, and HSC-3, see   [17, Section 6.1]) are stated in   [17,

Table 3.1,Table 6.1N] in the form of tabulated data that are illus-trated in Figs. 6 and 7.

The constitutive law of concrete in tension need not be defined

in our approach. It is sufficient to describe the tensile strength, f t ½Pa, that can be assumed as [15, Eq. (34)]

 f t ðhÞ¼ f t ;ref 

1 for   h6373:15 K;

ð873:15hÞ=500 for 373:15K <h6823:15 K;

ð1473:15hÞ=6500 for 823:15K <h61473:15 K;

0 for   h> 1473:15 K;

8>>><>>>:

  ð80Þ

Fig. 4.   Constitutive law for concrete in compression given by Eurocode 2   [17,Fig. 3.1].

Fig. 5.   An example of the constitutive law for concrete in compression at hightemperatures [17].

Fig. 6.  Reduction of the compressive strength of NSC and HSC given by Eurocode 2[17].

Fig. 7.  Temperature evolution of the mechanical strains of concrete according toEurocode 2 [17].

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where f t ;ref   ½Pa   is the reference tensile strength of concrete at the

room temperature.If the reference tensile strength, f t ;ref   ½Pa, is not known, it can be

estimated from the reference compressive strength of concrete atthe room temperature,   f c ;ref   ½MPa. In [16, Table 3.1],  the tensile

strength of high strength concrete is defined as

 f t ;ref  ¼ 2:12 ln 1 þ f c ;ref 

10   ½MPa:   ð81Þ

The Young’s modulus of concrete, Ec  ½Pa, can be easily derived from(79) in the form [see e.g. [18], Eq. (3.14)]

E c ðhÞ ¼   3 f c ðhÞ2c 1ðhÞ :   ð82Þ

The Poisson’s ratio of concrete, m ½—, is usually taken as temper-ature independent (see e.g.  [1, p. 46]) since the data about itstemperature dependency are ambiguous [49, Section 2.2.1.2]. Here,we follow the assumption (which leads to conservative results)that the Poisson’s ratio increases with increasing temperature.Based on the data experimentally determined by Mindeguia[44, pp. 132–133] for high strength concrete ( f c ;ref   60 MPa), we

assume that

mðhÞ ¼0:2 for   h6 293:15 K;

0:2 þ 0:5ðh 293:15Þ=580 for 293:15K < h6 873:15 K;

0:7 for   h > 873:15 K:

8><>:

  ð83ÞThe eccentricity, eF  ½—, used in the failure function of concrete

(Section 5.5) generally depends on the type of concrete and prob-ably also on temperature. Here, we assume constant (temperatureindependent) value,   eF  ¼ 0:505, derived from the data measuredby He and Song   [29, Table 2]   for high strength concrete( f c ;ref   60 MPa) under biaxial compression tests at high tempera-

tures, see  Fig. 8, where  f c ðhÞ   is taken from He and Song   [29,

Table 2],   f t ðhÞ   is assumed according to  (80), with   f t ;ref   ¼ 4 MPa,and F ðhÞ is determined by (26), with r3  ¼ 0.The porosity of concrete,  / ½—, may be expressed as  [21, Eq.

(41)]

/ðhÞ ¼ /ref  þ A/ðh href Þ;   ð84Þwhere / ref   ½—  is the reference porosity (at the reference tempera-

ture, href   ½K), and A/ ½K1  is a concrete-type-depend constant (see[21, pp. 46–47]).

The thermal conductivity of concrete,kc  ½W m1 K1, is given byGawin et al. [21, Eqs. (46) and (47)]

kc ðP ; hÞ ¼ kdðhÞ   1 þ 4 /ðhÞ qwðhÞ S wðP ; hÞð1 /ðhÞÞqs

;   ð85Þ

with

kdðhÞ ¼ kd;ref   1 þ Akðh href Þ

;   ð86Þ

where  kd;ref   ½W m1 K1  is the reference thermal conductivity of a

dry concrete (at the reference temperature,  h ref   ½K), and Ak  ½K1  isan experimentally determined coefficient.

For the intrinsic permeability of concrete,  K  ½m2, we adopt theBary function recommended by Davie et al. [14, Eq. (13)]

K ðP ; hÞ ¼ K ref   104DðP ;hÞ;   ð87Þ

where K ref   ½m2 is the reference permeability at the room tempera-ture, and D ½— is the multiplicative damage parameter that can bedefined as [13, Eq. (70)]; [24, Eq. (54)]

DðP ; hÞ ¼ DmðP ; hÞ þ DhðhÞ DmðP ; hÞDhðhÞ;   ð88Þwhere   Dm ½—   is the mechanical damage parameter, which isassumed to be equal to the failure parameter (31) in our approach,

i.e.

DmðP ; hÞ ¼ F ðP ; hÞ ð89Þand Dh ½— is the thermal damage parameter that we define as (cf.[24, Eq. (52)])

DhðhÞ ¼ 1 1

3  Dh;Ec  þ Dh; fc  þ Dh; ft 

¼ 1 1

3

E c ðhÞE c ;ref 

þ f c ðhÞ f c ;ref 

þ f t ðhÞ f t ;ref 

!:   ð90Þ

It should be noted that usually, the thermal damage parameteris based only on the degradation of the Young’s modulus (see [13,Eqs. (29–30)];   [24, Eq. (52)]). This is however not possible to

assume in our approach since we employ the constitutive modelproposed by Eurocode 2   [17], see   (79), where the stress isexpressed as a function of the total mechanical strain,  m, insteadof the instantaneous stress-related strain, r, and hence the result-ing softening of the material with increasing temperature is higherthan in reality (see also [2,28]). In order to eliminate this effect, wepropose to include not only the temperature dependent degrada-tion of  Ec  but also the reduction of the other mechanical properties( f c ; f t ), which leads to good agreement of the resulting thermaldamage parameter with the data stated in literature, see Fig. 9.

The gas relative permeability and the liquid water relative per-meability, K rg  ½— and K rw ½—, respectively, can be expressed by the

Fig. 8.   Comparison of failurefunction (26) used in our model with the experimentaldata measured by He and Song [29, Table 2].

Fig. 9.   Thermal damage assumed in our model (Dh) andits comparison with [13, Eq.(30)], Dh;½a, and [24, Eq. (52)], Dh;½b .

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formulas experimentally determined by Chung and Consolazio [10,Eqs. (7)–(9)]

K rg ðP ; hÞ ¼ 10S wðP ;hÞ  wðhÞ 10wðhÞ S wðP ; hÞ;   ð91Þ

K rwðP ; hÞ ¼ 10ð1S wðP ;hÞÞ  wðhÞ 10wðhÞ ð1 S wðP ; hÞÞ;   ð92Þwith

wðhÞ ¼ 0:05 22:5 /ðhÞ:   ð93ÞThe gas dynamic viscosity, l g  ½Pa s, is approximated as [21, Eq.

(50)]

l g ðP ; hÞ ¼ l g v ðhÞ þ ½l gaðhÞ l g v 

ðhÞ   P aP þ P a

0:608

;   ð94Þ

with

l g v ðhÞ ¼ l g v ;ref  þ av ðh href Þ ð95Þ

and

l gaðhÞ ¼ l ga;ref  þ aaðh href Þ þ baðh href Þ2;   ð96Þ

where   l g v ;ref   ¼ 8:85 106

Pa s;av  ¼ 3:53 108

P a s K1

,   l ga;ref   ¼17:17 106 Pa s;aa  ¼ 4:73 108 P a s K1;   ba  ¼ 2:22 1011 Pa s

K2, and  href   ¼ 273:15K.The liquid water dynamic viscosity, lw ½Pa s, can be calculated

from Gawin et al. [21, Eq. (51)]

lwðhÞ ¼ 0:6612ðh 229Þ1:532:   ð97Þ

The volume fraction of liquid (all free) water, gw ½—, is governedby the sorption isotherm function proposed in  [4–6]. After somemodification related to the saturated region,  P =P s P 1:0 (see [13,Eq. (73)]), we can write

if   h 6 hcr   :

gwðP ; hÞ ¼

c 

qwðhÞ/ðhref Þqwðhref Þ

c 

P 

P sðhÞ 1=mðhÞfor   P 

P sðhÞ6 0

:96

;P3i¼0niðhÞ   P 

P sðhÞ 0:96 i

for 0:96 <   P P sðhÞ <  1:00;

/ðhÞ   for   P P sðhÞP 1:00;

8>>>><>>>>:

if   h >  hcr   :

gw ¼ 0;

ð98Þwith

mðhÞ ¼ 1:04   ðh 263:15Þ2

22:34ðhref   263:15Þ2 þ ðh 263:15Þ2 ;   ð99Þ

where c ½kg m3 is the mass of cement per unit volume of concrete,

href   ½K   is the reference temperature (href   ¼ 298:15 K), the term½/ðhref Þqwðhref Þ   expresses the saturation water content at298:15 K, and   niðhÞ   are temperature dependent coefficients thatensure a smooth transition between the unsaturated and saturatedregion, thus the sorption isotherm function and its first derivativewith respect to   ðP =P sðhÞÞ   are continuous (cf.   [12,13,42,55]).Therefore, we can write

n0ðhÞ ¼ gw;0:96ðhÞ;   ð100Þn1ðhÞ ¼ g0

w;0:96ðhÞ;   ð101Þ

n2ðhÞ ¼ 3ðgw;1:00ðhÞ gw;0:96ðhÞÞ0:042

  2g0w;0:96ðhÞ þ g0

w;1:00ðhÞ0:04

  ;   ð102Þ

n3ðhÞ ¼ 2ðgw;0:96ðhÞ gw;1:00ðhÞÞ0:043

  þ g0w;0:96ðhÞ þ g0

w;1:00ðhÞ0:042

  ;   ð103Þ

where

gw;0:96ðhÞ ¼   c qwðhÞ

/ðhref Þqwðhref Þc 

  0:96 1=mðhÞ

;

gw;1:00ðhÞ ¼ /ðhÞð104Þ

and

g0w;0:96ðhÞ ¼ dgw;0:96ðhÞ

dh  and   g0

w;1:00ðhÞ ¼ dgw;1:00ðhÞdh

  :   ð105Þ

The degree of saturation with liquid water,  S w ½—, is defined as

S wðP ; hÞ ¼ gwðP ; hÞ/ðhÞ   :   ð106Þ

The mass of dehydrated water,   md ½kg m3, is governed byEq. (12), with [19, Eq. (8)]; [11, Eq. (C-22)]; [1, Eq. (1.9)]

md;eqðhÞ ¼   7:5

100m378:15

eq   1 exp   h 378:15

200

H ðh 378:15Þ

þ   2

100m378:15

eq   1 exp   h 673:15

10

H ðh 673:15Þ

þ  1:5

100m378:15

eq   1 exp   h 813:15

5

H ðh 813:15Þ;

ð107

Þwhere m378:15

eq   ½kg m3 is the equilibrium mass at 378:15 K and H  is

the Heaviside function.In [11, p. 146], the following values of material parameters that

belong to Eqs.   (12) and (107)   are stated:   s ¼ 10800 s;m378:15eq

¼ 210 kg m3, which are also adopted in our simulations, sincethese parameters are usually not measured within the fire tests.

The density of solid skeleton,  qs ½kg m3, is, together with theconcrete porosity end the mass of water released into the poresby dehydration, governed by the solid mass conservationEq. (A.8) (cf. [21, pp. 47–48]). In our approach, the density of solidskeleton is assumed to be a constant value. As obvious fromnumerical experiments, this simplification has negligible effecton the obtained results.

The density of liquid water, qw ½kg m3, can be expressed by theexperimentally determined formula, originally proposed by Fur-bish [20, Eq. (4.79)] and simplified by Gawin et al.   [23, Eq. (35)](neglecting the water pressure dependence of  qw), of the form

qwðhÞ ¼X5

i¼0

aiðh 273:15Þi 1 107

þX5

i¼0

biðh 273:15Þi for   h

6 hcr ;   ð108Þ

where   a0 ¼ 4:8863107; a1 ¼ 1:6528109; a2 ¼ 1:8621   1012;

a3 ¼ 2:42661013; a4 ¼ 1:59961015; a5 ¼ 3:3703   1018; b0 ¼

1:0213  103;   b1   ¼ 7:7377  101;   b2   ¼  8:7696  103;   b3   ¼

9:2118  105;   b4   ¼  3:3534  107;   b5   ¼ 4:4034  1010.

The specific enthalpy of evaporation,   he ½ J kg1, can be

expressed by the Watson formula   [21, Eq. (49)];   [26, Eq. (28)];[12, Eq. (AI.27)]

heðhÞ ¼   2:672 105ðhcr  hÞ0:38 for   h 6 hcr ;

0 for   h >  hcr :

(  ð109Þ

The specific enthalpy of dehydration, hd ½ J kg1, is considered to be

a constant value [12, Eq. (AI.26)]

hd ¼ 2400 103  J kg1

:   ð110Þ

The specific heat capacity of dry air, c a

 p  ½ J kg

1

K

1

, is determined byDavie et al. [12, Eq. (AI.28)]

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c a pðhÞ ¼ a h3 þ b h2 þ c  h þ d;   ð111Þ

where   a ¼ 9:84936701814735 108;   b ¼ 3:56436257769861

104;   c ¼ 1:21617923987757 101, and   d ¼ 1:0125025521632

4 103.

The specific heat capacity of liquidwater, c w p   ½ J kg1 K1, is given

by Davie et al.  [12, Eq. (AI.29)]

c w p ðhÞ ¼ ð2:4768 h þ 3368:2Þ þ   a h

513:15

b

for   h 6 hcr ;   ð112Þ

where a ¼ 1:08542631988638 and b ¼ 31:4447657616636.

The specific heat capacity of solid skeleton, c s p ½ J kg1 K1, canbe

calculated from Davie et al.  [12, Eq. (AI.30)]

c s pðhÞ ¼ 900 þ 80h 273:15

120  4

  h 273:15

120

2

:   ð113Þ

For the specific heat capacity of water vapour, c v 

 p   ½ J kg1 K1, the

following equation is proposed by Davie et al.  [12, Eq. (AI.31)]

c v 

 p ðhÞ ¼   ð7:1399 h 443Þ þ   a  h513:15

bfor   h 6 hcr ;

45821:01 for   h >  hcr ;

(  ð114Þ

where a ¼ 1:13771502228162 and b ¼ 29:4435287521143.It should be noted that above the critical temperature (if 

h >  hcr ), there is no liquid water in concrete (see Eq.  (98)) andhence, the liquid water properties (qw; c w p ) need not be defined.

9. Numerical results and experiments

In order to validate the model presented in this paper, theresults obtained by numerical simulations are compared with the

experimentally determined data. Three experiments reported inliterature are closely investigated:

 the test of the hygro-thermal behaviour of high strength con-crete ( f c ;ref   90 MPa) prismatic specimen 300 300

120 mm3 under unidirectional heating by the radiant heaterof a temperature of 600 C reported by Kalifa et al.   [36]  andrelated publications [35,37];

 the test of the hygro-thermal behaviour of high strengthconcrete ( f c ;ref   60 MPa) prismatic specimen 300 300

120 mm3 under unidirectional heating by the radiant heaterof a temperature of 600 C reported by Mindeguia   [44]   andrelated publications [45,47];

  the test of the spalling behaviour of high strength concrete

( f c ;ref   60 MPa) prismatic specimen 700 600 150 mm3

under unidirectional heating by the ISO 834 fire reported byMindeguia [44] and related publications [45,46].

Within the first two of the above tests, the pore pressure and thetemperature propagation through the specimen as well as its massloss were recorded. This type of tests is therefore denoted as the‘‘PTM test’’ by Mindeguia [44], which is also adopted in this paper.It should be noted that no spalling was observed during the bothPTM experiments and hence, these tests are employed primarilyfor the validation of the hygro-thermal behaviour simulation whilethe spalling prediction can be validated only on the qualitative level(whether it occurs or not). For that reason, the third test is alsoanalysed, since it enables to validate our model also with respect

to the quantitative prediction of spalling (i.e. the prediction of theamount of the potential spalling).

It should be noted that the usage of the model can provide onlyan approximate picture of the experiments described above. Theinaccuracies may arise mainly from the fact that:

 within the mechanical part of the model, a simplified constitu-tive law of concrete given by Eurocode 2 [17] is utilised. In thismodel, the transient strain is included implicitly and the creepstrain is neglected (see Section 5.3);

 in the simulation, the specimen is assumed to be fully mechan-ically restrained in the plane perpendicular to its heated sur-face. This assumption is however disputable in this cases. It isnot obvious, whether the ceramic blocks placed on the lateralsides of the small specimens within the PTM tests (seeFig. 10) or the unheated parts of the large specimens withinthe spalling test (see [44, Fig. 170]) provide a sufficient levelof restraint as assumed in the model (probably not);

 the model is not able to capture the ‘‘size effect’’ influencing thespalling behaviour of concrete specimens – i.e. that for smallspecimens, the spalling is less likely to occur in comparisonwith large specimens, as observed by recent experimentalworks [32,45].

9.1. Simulation of the PTM test 1

The Kalifa’s experiments  [35–37]   have been accepted as abenchmark problem by many authors focused on modelling of con-crete subjected to high temperatures (cf.   [1, Section III-3];   [13,Section 3.2];  [15, Section 5];  [22, Section 6.1];  [23, Section 6.2];[39, Sections 4.3,5.4]; [50, Section 6.4]; [57, Section 3.1]).

In [37], we can find the high temperature thermal and hygralproperties measured by Kalifa et al.   [37] for 4 types of concretenamed as M30, M75C, M75SC and M100C. The M30 and M100 con-cretes were used for the subsequent PTM tests reported by Kalifaet al. [36]. Within this experiment, the specimens were subjectedto an unidirectional heating up to 450 C, 600 C, and 800 C using

a radiant heater. Moreover, similar experiment – in this case forthe specimens made of C110 concrete (practically the same asthe M100 concrete, only with a little bit higher amount of cementand superplasticizer [35]) with polypropylene fibres of the contentof (0–3) kg m3, were made by [35] in order to assess the effect of fibres on the high-temperature behaviour of high strengthconcrete.

For the validation of our model, the results measured by Kalifaet al. [36] for the M100 concrete heated up to 600 C are employed.This case is chosen for the following reasons: our model is primar-ily designed to study the behaviour of high strength concrete (such

Fig. 10.   Scheme of the test set-up (according to 36, Fig. 2]).

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as M100); material properties of the M100 concrete are describedin detail in   [37]  (more comprehensive description than for theC110 concrete in   [35]); and the results obtained for the otherheating conditions (up to 450 C or 800 C) are not sufficientlyreported in [36].

9.1.1. Experiment descriptionThe tests performed by Kalifa et al. are described in detail in

[36]. Therefore, only a brief description of the experiment is statedbelow.

As mentioned above, two types of concrete (M30 and M100)and three different heating conditions (450 C, 600 C, and800 C) were used by Kalifa et al. [36]. Hereafter, only the case of M100 – 600 C is assumed. Material (thermal and hygral) proper-ties of the M100 concrete have been reported in   [37]   and arediscussed in Section 9.1.2.

As shown in Fig. 10, the 300 300 120 mm3 test specimen wassubjected to an unidirectional heating (in the 120-mm direction)by a radiant heater of the temperature of 600   C placed near itssurface for a heating period of 6 h.

Within the test, the temperature, the pore pressure, and the

mass loss of the specimen were recorded. The temperaturetogether with the pore pressure were measured using the com-bined pressure–temperature gauges (see   [35, Fig. 3]) placed atthe distance of (10, 20, 30, 40, and 50) mmfromthe heated surface.The heated surface temperature was measured by a thermocoupleplaced at the depth of 2 mm. Moreover, as obvious from the graphslisted in   [36], the temperature on the unexposed side was alsorecorded (probably by an additional thermocouple or by a surfacethermometer). The mass loss was monitored by a balance on whichthe specimen was placed during the test (see Fig. 10).

9.1.2. Modelling In this section, the material properties of concrete, the initial

and boundary conditions as well as the discretization employedfor the numerical modelling are discussed. The thermal and hygralproperties of concrete are assumed according to the relationshipsgiven in Section 8, with the parameters determined from the datameasured by Kalifa et al. [37]. All the parameters are summarisedin Table 1.

The concrete of the test specimens (named as M100 concrete in[36,37]) is assumed to be a high strength concrete, class 2 (see [17,Section 6.1] and Section 8), with calcareous aggregates, which canbe denoted as ‘‘HSC2-C’’. This classification determines the freethermal strain of concrete and the temperature dependent reduc-

tion of its compressive strength according to  [17], as described inSection 8.

The reference compressive strength of concrete at the roomtemperature is taken from Kalifa et al. [37, Table 1]. The referencetensile strength of concrete at the room temperature is notspecified in [36,37] and hence, it is calculated by (81).

The mass of cement per unit volume of concrete is taken fromKalifa et al. [37, Table 1].

For the porosity of concrete, the parameters of Eq.  (84)   aredetermined by a linear regression of the data stated in   [37,Table 2], see Fig. 11.

Assuming the above values of  c  and  /ðhÞ, the resulting degreeof saturation with liquid water   (106), based on the sorptionisotherms (98), is illustrated in Fig. 12.

As mentioned in Section 8, the density of solid skeleton,  qs, isassumed to be a constant value in our approach. This value canbe estimated from the data measured by Kalifa et al.   [37,Table 4]   for the apparent density. The corresponding apparentdensity assumed in our model, that can be expressed as (cf.  [24,Eq. (26)])

q

¼qw gw

þq g  g g 

þqs gs

¼ qwð/ S wÞ þ ðqv  þ qaÞ½/ð1 S wÞ þ qsð1 /Þ;   ð115Þ

appears in Fig. 13.The temperature dependence of the thermal conductivity of 

concrete was measured by Kalifa et al. [37, Table 7] at a dry state.The parameters of Eq. (86) are determined by a linear regression of the measured data, see Fig. 14 (RH ¼ 0). The resulting thermal con-ductivity of concrete assumed in our model is shown in  Fig. 14.

The intrinsic permeability of concrete was tested by Kalifa et al.[37, Table 6]   on concrete samples dried at 105 C and thenheated up to several temperature levels. However, the reference

 Table 1

Material properties and parameters used in our simulation of the PTM test 1.

Parameter Value Unit Reference

Type of concrete HSC2-C –   [17];[37]

 f c ;ref    91.8 MPa   [37, Table 1]

 f t ;ref    4.9 MPa   [16], see (81)

c    414.8   kg m3 [37, Table 1]

href    293.15 K Determined from

/ref    0.0897 –   [37, Table 2],

 A/   2:4457 105 K1 see Fig. 11

qs   2660   kg m3 Determined from

[37, Table 4],see Fig. 13

href    293.15 K Determined from

kd;ref    1.9759   W m1 K1 [37, Table 7],

 Ak   6:4215 104 K1 see Fig. 14

K ref    1:3 1020 m2 –

Fig. 11.  Porosity of concrete measured by Kalifa et al. [37] (points) and assumed inour model (line).

Fig. 12.  Degree of saturation with liquid water assumed in our model.

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permeability of undamaged concrete at the room temperature isnot reported in [37] and hence it is determined by a trial-and-errormethod in our simulation (cf.  [57, p. 277]).

Scheme of the analysed problem is displayed in Fig. 15. The spa-tial discretization is performed with the use of linear 1-D elements.

In total, 120 elements are employed – 30 elements in the interval x 2 ð0; ‘=2Þ, 30 elements for x 2 ð‘=2; 3‘=4Þ and 60 elements in theinterval x 2 ð3‘=4; ‘Þ. For the time discretization, the time step isset to   Dt ¼ 1 s. The characteristic time of spalling is assumed asc ¼ 10 s.

The initial conditions are assumed as  P 0  ¼ 1:9039 103 Pa;  h0

¼ 293:15 K; ‘0  ¼ 0:12 m. For these values, the initial saturationwith liquid water S w;0  ¼ 0:77, which is the value reported by Kalifaet al. [36, Table 2].

As stated above, a radiant heater of a temperature of 600 C was

employed for the experiment. However, as mentioned by e.g. [23,pp. 554–556], the temperature of the ambient air on the heatedside, which needs to be defined for the boundary conditionsassumed in the model (see Section 4), is lower than the tempera-ture of the heater. For our simulation, the ambient temperatureon the heated side, h1 ½K, has been determined by a trial-and-errormethod as (see Fig. 16)

h1ðt Þ ¼   293:15 þ t 410300

  for   t 6 300 s;

703:15 þ ðt  300Þ   3521300

  for   t  >  300 s;

(  ð116Þ

where t ½s is the time of heating.The boundary conditions are summarised in Table 2.

9.1.3. DiscussionThe resulting pore pressure, temperature and mass loss

evolutions determined by our simulation are compared with theexperimentally measured data in Figs. 17–19 (note that the dis-tances in millimeters stated in the figures are, in accordance with[36], measured from the heated surface).

The pore pressure evolution in the depth of 30 mm from theheated surface is not displayed in Fig. 17 since it was not correctlymeasured by Kalifa et al.  [36], probably due to the damage of thegauge (see [36, p. 1923]). On the other hand, the pore pressureevolution in the depths of (10, 20, and 50) mm can be verified moreprecisely because the investigated PTM test was duplicated and thedata from the both measurements are stated in  [36, Fig. 6], see

Figs. 20–22.From Figs. 17–22, it is obvious that in this case, despite of the

simplifications mentioned in the introductory part of this section,the present model predicts the hygro-thermal behaviour of theheated concrete specimen on the sufficient level of accuracy. Itshould be however mentioned that some of the potential inaccura-cies arising from the simplifications embedded in the model areprobably eliminated by the appropriate setting of the referencepermeability of undamaged concrete at the room temperature(fitted by a trial-and-error method).

The spalling behaviour is also simulated correctly since, as men-tioned above, during the test, no spalling was observed by Kalifaet al.   [36], which is in accordance with our simulation, seeFig. 23 – the maximal value of failure function (31) achieved within

the specimen during the test period does not exceeds the value of 1. The evolution of the maximal values of the damage parameter

Fig. 14.  Thermal conductivity of concrete measured by Kalifa et al.  [37] at a drystate (points) and assumed in our model (lines).

Fig. 16.   Air temperature on the heated side of the specimen used for the simulationof the PTM test 1.

Fig. 15.  Scheme of the analysed problem.

Fig. 13.   Apparent density of concrete measured by Kalifa et al. [37] at a dry state(points) and assumed in our model (lines).

 Table 2

Boundary conditions parameters for simulation of PTM test 1.

Variable Value Unit

Unexposed side Exposed side

P 1   P 0   P 0   Pah1   h0   h1ðt Þ, see (116)   Kac    4 20   W m2 K1

erSB   0:7 5:67 108 0:7 5:67 108 W m2 K4

bc    0:009 0:019   m s1

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and its components within the specimen during the test perioddetermined by our model is shown in Fig. 24.

For illustration, the spatial and time distributions of the primaryunknowns of the present model (except the thickness of the

specimen, which is constant in this case) for the Kalifa’s PTM test1 [36] are shown in Figs. 25 and 26.

9.2. Simulation of the PTM test 2

9.2.1. Experiment descriptionThe PTM test procedure proposed by Kalifa et al.   [36]   was

adopted by Mindeguia as a part of an extended experimental pro-gram reported in [44–47]. Within this program, the material prop-erties investigation, hygro-thermal behaviour measurements (thePTM tests) and the spalling experiments were performed (adetailed summary of the experiments included in the programappears in [44, Section 1]). Hereafter, one of the PTM tests per-formed by Mindeguia [44] is closely investigated – the PTM testof a high strength concrete (denoted as ‘‘B60’’ in [44]) prismaticspecimens (conditioned in ‘‘Air_1’’ according to [44]) heated by aradiant heater of the temperature of 600 C (the type of heatingdenoted as ‘‘modéré’’ in [44]) placed near its surface for a heatingperiod of 5 h. The test set-up is the same as described for the Kali-fa’s experiment (Section 9.1.1). Note that (i) within the investi-gated PTM test 2, no spalling was observed by Mindeguia   [44],

and (ii) the test was duplicated and the data from the both mea-surements of  P  are stated in [44], see Section 9.2.3.

Fig. 17.  Pore pressure evolution measured by Kalifa et al.  [36, Fig. 4] (dashed lines) and determined by the present model (solid lines).

Fig. 18.  Temperature evolution measured by Kalifa et al.  [36, Fig. 4] (dashed lines) and determined by the present model (solid lines).

Fig. 19.   Mass loss measured by Kalifa et al. [36, Fig. 7] (dashed line) and determinedby the present model (solid line).

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9.2.2. Modelling The thermal and hygral properties of concrete are assumed

according to the relationships given in Section 8, with the param-eters determined from the data measured by Mindeguia [44]. Allthe parameters are summarised in Table 3.

The concrete of the test specimens (denoted as C60 concrete in[44]) is assumed to be a high strength concrete, class 1 (see  [17,Section 6.1]  and Section   8), with calcareous aggregates, whichcan be denoted as ‘‘HSC1-C’’. This classification determines the freethermal strain of concrete and the temperature dependent reduc-tion of its compressive strength according to  [17], as described inSection 8.

The reference compressive and tensile strengths of concrete atthe room temperature are taken from Mindeguia [44, Table 24].

The mass of cement per unit volume of concrete is taken fromMindeguia [44, Table 14].

For the porosity of concrete, the parameters of Eq.  (84)   aredetermined by a linear regression of the data stated in   [44,

Table 24], see Fig. 27.The density of solid skeleton,  qs, is estimated from the data

measured by Mindeguia [44, Table 26]. The corresponding appar-ent density assumed in our model appears in  Fig. 28.

The parameters of Eq. (86) for the thermal conductivity of dryconcrete are determined by a linear regression of the data statedin   [44, Table 28]  (excluding the value for the temperature of 20 C), see Fig. 29 (RH ¼ 0). The resulting thermal conductivity of concrete assumed in our model is shown in  Fig. 29.

The intrinsic permeability of concrete was tested by Mindeguia[44] on concrete samples dried at 80 C  and then heated up to sev-eral temperature levels. However, the reference permeability of undamaged concrete at the room temperature is not reported in[44]   and hence it is determined by a trial-and-error method in

our simulation (cf. [57, p. 277]).The scheme of the analysed problem, the spatial discretization,

the time discretization, and the characteristic time of spalling areidentical as in the previous example (see Section  9.1.2).

The initial conditions are assumed as

P 0  ¼ 1:9194 103 Pa;h0  ¼ 293:15 K;‘0  ¼ 0:12 m. For these values,the initial saturation with liquid water  S w;0  ¼ 0:78, which is thevalue reported by Mindeguia [44, Table 18].

The ambient temperature on the heated side,  h1 ½K, has beendetermined by a trial-and-error method as (see Fig. 30)

h1ðt Þ ¼ 293:15 þ t 380300

  for   t 6 300 s;

673:15 þ ðt  300Þ   5017;700

  for   t  >  300 s;

(  ð117Þ

where t ½s is the time of heating, and it is practically the same as forthe Kalifa’s PTM test 1, cf. Figs. 16 and 30.

Fig. 20.   Pore pressure development measured by Kalifa et al.   [36, Fig. 6]   anddetermined by the present model at the depth of 10 mm from the heated surface.

Fig. 21.   Pore pressure development measured by Kalifa et al.   [36, Fig. 6]   anddetermined by the present model at the depth of 20 mm from the heated surface.

Fig. 22.   Pore pressure development measured by Kalifa et al.   [36, Fig. 6]   and

determined by the present model at the depth of 50 mm from the heated surface.

Fig. 23.  Evolution of the failure parameter.

Fig. 24.  Evolution of the damage parameter.

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The boundary conditions are summarised in Table 4.

9.2.3. DiscussionThe resulting pore pressure and temperature evolutions deter-

mined by our simulation are compared with the experimentallymeasured data in Figs. 31 and 32 (note that the distances in milli-

meters stated in the figures are, in accordance with [44], measuredfrom the heated surface). The more detailed illustration of the pore

pressure prediction and its comparison with the test data is shownin Fig. 33.

From Figs. 31–33, it is obvious that also in this case, the presentmodel provides an accurate prediction of the hygro-thermalbehaviour of the heated concrete specimen. Similarly as in the pre-vious case, the potential inaccuracies arising from the simplifica-tions of the model (see the introductory part of this section) areprobably excluded by the appropriate setting of the reference

Fig. 25.   Distribution of  P  and  h  for the analysed PTM test 1.

Fig. 26.   Distribution of  m  and md  for the analysed PTM test 1.

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permeability of undamaged concrete at the room temperature (fit-ted by a trial-and-error method).

The spalling behaviour is also simulated correctly since, as men-tioned above, during the test, no spalling was observed by Minde-guia [44], which is in accordance with our simulation, see Fig. 34 –the maximal value of failure function   (31)  achieved within thespecimen during the test period does not exceeds the value of 1.The evolution of the maximal values of the damage parameterand its components within the specimen during the test perioddetermined by our model is shown in Fig. 35.

9.3. Simulation of the spalling test 

9.3.1. Experiment descriptionThe spalling experiments were performed by Mindeguia [44] onthe 700 600 150 mm3 prismatic specimens made of varioustypes of concrete and exposed to various fire scenarios. Hereafter,the spalling test of a high strength concrete (denoted as ‘‘B60’’ in[44]) sample (conditioned in ‘‘Air_2’’ according to [44]) heated bythe ISO 834 fire (in the 150-mm direction) for 60 min. This testwas duplicated and the data from the both measurements of thespalling depths are stated in   [44]. The specimen was placed onthe top of the furnace and it was exposed to heating on the areaof 420 600 mm2 (see [44, Fig. 170]). As mentioned by Mindeguia[44] and Mindeguia et al. [46], during the test, the specimen wasnot subjected to any mechanical load.

Within the test, the temperature and the pore pressure propa-

gations through the specimen were recorded. After the terminationof heating, the spalling depths on the exposed surface were closely

measured which enables to determine the maximal and average

depths of spalling and depict the well-arranged ‘‘spalling maps’’(see [44,46]).

 Table 3

Material properties and parameters used in our simulation of the PTM test 2.

Parameter Value Unit Reference

Type of concrete HSC1-C –   [17];[44]

 f c ;ref    61:0 MPa   [44, Table 24]

 f t ;ref    3:76 MPa

c    550   kg m3 [44, Table 14]

href    293:15 K Determined from/ref    0:1027 –   [44, Table 25],

 A/   1:0624 104 K1 see Fig. 27

qs   2660   kg m3 Determined from

[44, Table 26],see Fig. 28

href    293.15 K Determined from

kd;ref    2.0153   W m1 K1 [44, Table 28],

 Ak   9:8533 104 K1 see Fig. 29

K ref    4:0 1020 m2 –

Fig. 28.  Apparent density of concrete measured by Mindeguia  [44]  (points) andassumed in our model (lines).

Fig. 29.   Thermal conductivityof concrete measured by Mindeguia [44] (points) andassumed in our model (lines).

Fig. 30.   Air temperature on the heated side of the specimen used for the simulation

of the PTM test 2.

Fig. 27.  Porosity of concrete measured by Mindeguia [44] (points) and assumed inour model (line).

 Table 4

Boundary conditions parameters for simulation of PTM test 2.

Variable Value Unit

Unexposed side Exposed side

P 1   P 0   P 0   Pah1   h0   h1ðt Þ, see (117)   Kac    4 20   W m2 K1

er   0:7 5:67 108 0:7 5:67 108 W m2 K4

bc    0:009 0:019   m s1

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The detailed description of the spalling experiments performedby Mindeguia can be found in [44,46].

9.3.2. Modelling The thermal and hygral properties of the B60 concrete as well as

its hygro-thermal behaviour are described in Section 9.2.2.Scheme of the analysed problem is displayed in Fig. 36. The spa-

tial discretization is performed with the use of linear 1-D elements.In total, 160 elements are employed – 40 elements in the interval x 2 ð0; ‘=2Þ, 40 elements for x 2 ð‘=2; 3‘=4Þ and 80 elements in theinterval   x 2 ð3‘=4; ‘Þ. For the time discretization as well for thecharacteristic time of spalling, three values are assumed:Dt  ¼ 1 s;  Dt ¼ 0:5 s;  Dt  ¼ 0:1 s, and   c ¼ 1 s;  c ¼ 10 s , andc ¼ 100 s, respectively, which leads to 9 numerical experimentsdiscussed in Section 9.3.3.

The initial conditions are assumed as   P 0  ¼ 1:9194 103 Pa;

h0  ¼ 293:15 K; ‘0  ¼ 0:15 m. For these values, the initial saturationwith liquid water  S w;0  ¼ 0:78, which is the average value of thetwo initial saturations reported by Mindeguia [44, Table 20].

The ambient temperature on the heated side is governed by theISO 834 fire curve

h1ðt Þ ¼ 293:15 þ 345logð8t þ 1Þ;   ð118Þ

where t ½minutes is the time of heating.The boundary conditions are summarised in Table 5.

9.3.3. DiscussionIn this case, the spalling of concrete occurred during the test,

which is also correctly simulated by the present model. The reduc-tion of the specimen thickness obtained by the calculations is

shown in Fig. 37.The spatial and time distributions of the primary unknowns of 

the present model are shown in Fig. 38.It is obvious that the model predicted a large amount of con-

crete spalling – the spalling depth at the end of heating reachesthe values from 85 mm to 99 mm, depending onDt  and c assumedfor the calculation. These values of spalling depths are about 2.5 to4 times as high as measured by Mindeguia [44, Table 36] the max-imal spalling depths of 34 mm and 23 mm for the first and for thesecond test are reported, respectively.

Moreover, the values of pore pressure determined by the simu-lation are also significantly higher than measured within theexperiment. The maximal value of pore pressure obtained by thesimulation is about 2:5 MPa (see Fig. 38); the maximal value of 

pore pressure measured by Mindeguia is 0:63 MPa (see   [44,Table 38]).

Fig. 31.  Pore pressure evolution measured by Mindeguia [44, Fig. 154c] (dashed lines, dotted lines) and determined by the present model (solid lines).

Fig. 32.  Temperature evolution measured by Mindeguia [44, Fig. 157c] (dashed lines) and determined by the present model (solid lines).

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The overestimation of pore pressures indicates that in such arapid heating conditions, the specimen was damaged more

severely (which leads to increase of the permeability of concrete)

than has been predicted by the model. The fact that the increasingheating rate leads to decreasing pore pressures within the space-men was observed by Mindeguia et al. [45, Section 3.2]. In our sim-ulations, this trend is not so evident. When comparing the porepressure distributions within the specimen heated by a radiantheater of a temperature of 600 C (see Figs. 31, 33) with the porepressure in the specimen made of the same concrete exposed tothe ISO 834 fire (see Fig. 38), we get practically identical values(about 2:5 MPa). In our opinion, there are three possible explana-

tions of this fact:

Fig. 34.  Evolution of the failure parameter.

Fig. 33.  Pore pressure evolution measured by Mindeguia [44, Fig. 157c] and determined by the present model at the depth of (10, 20, 40, 50) mm from the heated surface.

Fig. 35.  Evolution of the damage parameter.

 Table 5

Boundary conditions parameters for simulation of spalling test.

Variable Value Unit

Unexposed side Exposed side

P 1   P 0   P 0   Pah1   h0   h1ðt Þ, see (118)   Kac    4 25   W m2 K1

er   0:7 5:67 108 0:7 5:67 108 W m2 K1

bc    0:009 0:019   m s1

Fig. 36.  Scheme of the analysed problem.

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Fig. 37.  Reduction of the specimen thickness due to concrete spalling for the analysed experiment  [44] obtained by the present model.

Fig. 38.  Spatial and time distribution of the primary unknowns for the analysed experiment  [44] obtained by the present model (for  Dt  ¼ 0:1 s and c ¼ 1 s).

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 the Bary function (87) adopted in our model from Davie et al.[14, Eq. (13)] is not valid for such a rapid heating conditionsas the ISO 834 fire exposure;

  the mechanical damage parameter influencing the increase of concrete permeability cannot be simply assumed to be equalto the failure parameter assumed in our spalling criterion, aswe supposed in Eq. (89);

 the value of reference permeability of undamaged concrete atthe room temperature that has been set in our simulation of the spalling test is not correct. It has been adopted from the pre-vious PTM test of small specimen made of the same concrete.Within the PTM test, concrete permeability has been fitted bya trial-and error method and hence, it is possible that it coversindirectly some of the factors related to the PTM experiment of small samples. However, these factors do not need to be rele-vant for the spalling test of large specimens.

The detailed investigation of this phenomenon needs furtherresearch and it is out of scope of the present paper.

The overestimation of the specimen thickness reduction isconsequently caused by the overestimation of pore pressures asdiscussed in the previous paragraph, and probably also by theassumption adopted in the model that the specimen is fullymechanically restrained in the plane perpendicular to its thick-ness. In the investigated experiment, the only restraint arose fromthe fixing the heated concrete by the unheated lateral parts andupper layers of the specimen (see [32] and references therein). Itshould be noted that the overestimation of the spalling processhas been supposed since the assumption of the fully restrainedconditions is generally considered as conservative. Moreover, inthe real conditions, the structure usually is mechanicallyrestrained and hence, in such a case, the model can provide moreaccurate prediction.

Nevertheless, numerical experiments demonstrate a computa-tional stability of the present model. From Fig. 37, it is obvious thatthe model is not extremely sensitive on the value of time step. It is

also evident that by increasing the characteristic time of spalling aswell as by decreing the time step, the spalling process becomessmoother.

10. Conclusions

In the paper, we proposed a one-dimensional coupled model forsimulation of hygro-thermo-mechanical behaviour of concretewalls exposed to high temperatures including the prediction of concrete spalling. The transport-processes-related part of themodel was derived from the multi-phase formulations of the lawsof conservation of mass and energy. Based on the detailed quanti-tative parameter analysis, the less important transport phenomenawere neglected in order to obtain a simplified but still robust and

realistic model of transport processes in heated concrete, whichcan be expressed in terms of relatively small number of materialparameters that can be easily obtained from material tests. Themain simplifications employed in our approach consisted in (i)neglecting the diffusive mass flux of water vapour, (ii) removingthe separate term describing the effect of diffusion of adsorbedwater and including this effect into the liquid water relative per-meability, K rw, and (iii) ignoring the effects of variations of pressureof dry air. The permeability of concrete was supposed to be influ-enced both by the thermal and mechanical damage of the material,which can be expressed by a multiplicative total damageparameter.

The spalling of concrete was assumed to be caused by a combi-nation of the hygro-thermal stress due to the pore pressure build-

up and the thermo-mechanical stress resulting from the restrained

thermal dilatation. We employed a simplified mechanicalapproach based on the assumption that the wall is fully mechani-cally restrained and the effects of stresses resulting from the exter-nal mechanical load as well as the self-weight of the wall on thepotential spalling are neglected. An evolution law for movingboundary due to spalling was proposed. It enables to simulatethe instantaneous spalling of concrete as a continuous process,which contributes to the computational stability of the numericalsolution.

For the model represented by a system of partial differentialequations and appropriate boundary and initial conditions, thefinite element discretization in space and the semi-implicitdiscretization in time were employed. The resulting numericalalgorithm was incorporated into an in-house MATLAB code thatwas applied for numerical experiments in order to validate thepresent model.

The validity of the model was verified by comparing the resultsobtained by the numerical simulations with the data measuredwithin real high-temperature experiments. Two types of tests wereinvestigated – the tests of hygro-thermal behaviour of high-strength concrete specimens 300 300 120 mm3 heated by aradiant heater of temperature of 600 C performed by Kalifa et al.[36], Mindeguia [44] and the spalling test of high-strength concretespecimens 700 600 150 mm3 exposed to the ISO 834 fire per-formed by Mindeguia   [44]. All the material properties assumedfor the simulations, except the reference intrinsic permeability of undamaged concrete at the room temperature,  K ref , were adoptedfrom the data measured by Kalifa et al.  [36] and Mindeguia [44],respectively. The values of  K ref   were not reported by the authorsof the tests ([36,44] reported only the values of the intrinsic per-meability dried at 105 C or 80 C, respectively, and then heatedup to several temperature levels).

By comparing the calculated and measured data, the followingfindings can be drawn:

  the model is able to simulate the hygro-thermal behaviour of heated concrete on an appropriate level of accuracy, the tem-perature and pore pressure distributions obtained by the pres-ent model are in close agreement with the data measured byKalifa et al. [36] and Mindeguia [44];

  for the investigated examples, the model provided a correctqualitative prediction of concrete spalling (whether it occurredor not);

 the quantitative prediction of spalling for the Mindeguia’sspalling test   [44]   was overestimated (the values of spallingdepths determined by the model at the end of the test periodare about 2.5 to 4 times as high as measured by Mindeguia[44]). This result arises from the overestimation of pore pres-sures (as discussed in detail in Section 9.3.3) and probably alsofrom the assumption of the full mechanical restraint of the

specimen. This assumption has been adopted in the modelbut it is probably not fulfilled in the investigated test setupby Mindeguia [44]. It can be supposed that in real structures(with some level of restraint), the model can provide moreaccurate prediction.

For future research, it could be recommended to focuss on:

 determination of the reference intrinsic permeability of undam-aged concrete at the room temperature;

 validation of the present model on a real structure exposed tofire or on the results of full-scale fire experiments;

 extension of the present model for two-dimensional problemsand improvement of the mechanical part of the model in order

to describe various types of loading and restrain conditions.

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Conflict of interest

None declared.

 Acknowledgements

This research has been supported by the Czech Science Founda-

tion, project GAC R 13–18652S (the first and the second author),and by the Grant Agency of the Czech Technical University inPrague, project SGS13/038/ OHK1/1T/11 (the second author). Thesupport is gratefully acknowledged.

 Appendix A. Physical mechanisms of transport processes in

concrete at high temperatures

In this Appendix we address, on the quantitative level, contribu-tions of different transport mechanisms in concrete exposed toambient high-temperatures. To this goal, we will first summarisemodelling assumptions and state variables employed in the deriva-tion of complete multi-phase formulations, together with thedependence of material constants on the state variables. Then, by

expressing the thermodynamic fluxes in terms of the same statevariables, we will obtain closed-form expressions for the corre-sponding transport coefficients. Their relative importance will beevaluated at the level of material point and these results will allowus to explore the domains of applicability and the connection of the presented simplified model against detailed multi-phasedescriptions.

 A.1. Conservation of mass

The local form of the macroscopic mass balance equation of thephase a  is [24]

Daqa

Dt   þ qar v a ¼ Ma:   ðA:1Þ

Here,

qa ¼ gaqa   ðA:2Þrepresents the phase averaged density, ga [–] is the volume fractionof the a  phase and qa  stands for the intrinsic phase averaged den-sity. Note thatXa

ga ¼ 1:   ðA:3Þ

Further, v a [m s1] is the velocity of a phase and Ma is the volumet-ric mass source. Using (A.2) Eq. (A.1) can be rewritten as

@ ðgaqaÞ@ t 

  þ r   gaqav að Þ ¼ Ma:   ðA:4Þ

Based on multi-phase modelling, the moist concrete is consid-ered as a multi-phase porous material consisted of solid skeletonwith pores filled by liquid water and gas, which is a mixture of dry air and water vapour. Let us write the balance Eq. (A.4) for eachphase of concrete mixture:

Liquid water conservation equation:

@ ðgwqwÞ@ t 

  þ r   gwqwv wð Þ ¼ @ me

@ t  þ @ md

@ t   ;   ðA:5Þ

Water vapour conservation equation:

@ ðgv q

v Þ

@ t   þ r   g

v q

v v v 

¼ @ me

@ t   ;   ðA:6Þ

Dry air conservation equation:

@ ðgaqa

Þ@ t    þ r   gaqav að Þ ¼ 0;   ðA:7Þ

Solid mass conservation equation:

@ ðgsqsÞ@ t 

  þ r   gsqsv sð Þ ¼ @ md

@ t   ;   ðA:8Þ

where md  [kg m3] is the mass source term related to the dehydra-tion process and me  [kg m3] is the vapour mass source caused bythe liquid water evaporation.

The mass balances of the liquid water and of the vapour,summed together to eliminate the source term related to phasechanges (evaporation or condensation), form the mass balanceequation of moisture (liquid water and vapour):

@ 

@ t ðgwqw þ g

v q

v Þ þ r   gwqwv w þ g

v q

v v v 

¼ @ md

@ t   :   ðA:9Þ

 A.2. Moisture flux

The total moisture flux in moist concrete can be expressed as

 J M  ¼ gwqwv w |fflfflfflfflffl{zfflfflfflfflffl}  J w

þg g qv v v  |fflfflfflffl{zfflfflfflffl} 

 J v 

;   ðA:10Þ

where J w represents the liquid (capillary and adsorbed) water massflux and J v  is the water vapour mass flux.

The chosen primary state variables are temperature   h, watervapour pressure P v  and dry air pressure P a. We express the ther-modynamic fluxes in terms of state variables  h; P v  and P a  to obtainthe corresponding transport coefficients related to differentphysical mechanisms of moisture transfer.

Liquid water mass flux.  Different mechanisms governing theliquid (free) water mass flux need to be distinguished. In particular,above the solid saturation point the transport of liquid water con-sists of capillary water flows driven by the capillary pressure gra-dient. Otherwise, the physically adsorbed water flows diffusesdue to the saturation gradient. Consequently, mathematicalexpression of liquid water (adsorbed or capillary) mass flux readsas (cf. [12])

 J w ¼   1 S BS w

qw

KK rw

lw

rðP  g   P c Þ S BS w

qwDBrS B;   ðA:11Þ

where the degree of saturation with adsorbed water   S B   isdefined as

S B ¼S w   for   S w  6 S ssp;

S ssp   for   S w >  S ssp;

8<: ðA:12Þ

S w  represents the degree of saturation with liquid water and  S ssp  isthe solid saturation point.  Db  is the bound water diffusion tensor(see [21, Eq. (39)]).

Note that the saturation degree with liquid water S w is definedas

S w ¼ gw

/  ½:   ðA:13Þ

Here the volume fraction of liquid water   gw   in concrete can becalculated via the sorption isotherms (introduced by Baz ant et al.[4–6]) as a function of temperature  h  and vapour pressure P v .

The equilibrium state of the capillary water with the watervapour is expressed in the form corresponding to the Kelvinequation

P c ðP v ; hÞ ¼ qw

Rh

M wln

  P v 

P s

;   ðA:14Þ

where P c  denotes the capillary pressure and the water vapour satu-ration pressure P s can be calculated from the following formula as a

function of temperature  h  [K]

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P sðhÞ ¼ exp 23:5771   4042:9

h 37:58

:   ðA:15Þ

Finally, the liquid water mass fluxes can be expressedin the closed-forms with respect to gradients of primary statevariables:

Adsorbed water diffusion (S w  6 S ssp; S B ¼ S w):

gwqwv w  ¼ qwDBrS w

¼ qwDB

@ S w@ P v 

rP v  qwDB

@ S w@ h

 rh;

ðA:16Þ

Capillary water flow (S w  >  S ssp; S B ¼ S ssp):

gwqwv w ¼   1 S BS w

qw

KK rw

lw

rP  g þ   1 S BS w

qw

KK rw

lw

rP c 

¼   1 S BS w

qw

KK rw

lw

rP  g þ   1 S BS w

qw

KK rw

lw

@ P c 

@ P v 

rP v þ@ P c 

@ hrh

¼   1 S BS w

  qw

KK rw

lw

þqw

KK rw

lw

@ P c 

@ P v 

rP v 

  1

S B

S w qw

KK rw

lw rP a

þ  1

S B

S w qw

KK rw

lw

@ P c 

@ h rh:

ðA:17ÞWater vapour mass flux.  Decomposition of the water vapour

velocity into the diffusional (v v   v  g ) and advectional (v  g ) compo-nents yields

Fig. 39.  Moisture fluxes related to rP v  representing the different moisture transfer mechanisms – (I) vapour flow, (II) vapour diffusion, (III) liquid water flow, (IV) adsorbedwater diffusion.

Fig. 40.  Moisture fluxes related to rP v  representing the different moisture transfer mechanisms – (I) vapour flow, (II) vapour diffusion, (III) liquid water flow, (IV) adsorbedwater diffusion.

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 J v  ¼ g g qv 

v  g  þ g g qv ðv v   v  g Þ:   ðA:18Þ

The advective flux may be described by Darcy’s law in the form(vapour flow)

g g qv v  g  ¼ q

v 

KK rg 

l g 

rP  g 

¼ q

v 

KK rg 

l g  rP v 

 q

v 

KK rg 

l g  rP a:

ðA:19Þ

For the diffusive mass flux of water vapour, Fick’s law (vapour dif-fusion) is applied in the form

 J v 

d ¼ g g qv ðv v   v  g Þ ¼ q g 

M aM w

M 2 g 

Deff r   P v 

P  g 

¼   M aM wP ahRðP v M w þ P aM aÞ Deff rP v 

þ   M aM wP v 

hRðP v M w þ P aM aÞ Deff rP a:

ðA:20Þ

Total moisture flux. The total moistureflux can be expressed in termsof gradients of state variables P v ; P a  and  h  to obtain

Figs. 39 and 40 present the dependence of transport coefficientsupon the temperature and relative humidity at constant atmo-spheric pressure (P a ¼ 101325 Pa) related to the gradient of vapourpressure   rP v . Material properties of concrete are taken fromGawin et al. [21]. The moisture flux according to gradient of vapourpressure, as derived in  (A.21), consists of Darcian vapour flow,vapour diffusion and Darcian flow of capillary water (providedS w  >  S ssp) or diffusion of adsorbed water (provided   S w  6 S ssp).Figs. 39 and 40  show that with increasing temperature Darcianvapour flow plays a dominant role (approximately above 200 C).This observation is in accordance with [15] where it is assumed

that only the vapour transport takes place during the temperatureexposure of HSC. At lower temperatures, the adsorbed water diffu-sion or Darcian water flow (depending on degree of saturation) areof importance. Moreover, at the temperatures between100 200 C in the relatively small  RH -range, approximately upto 10%, the vapour diffusion may be the dominant factor of mois-ture transport. However, following our observation based onnumerical experiments, neglecting vapour diffusion has negligibleeffects on the spatial position of the amplitude of the peak and thepeak value of the pore pressure.

In order to simplify the model as much as possible andminimise the number of input parameters, against completemulti-phase formulation, where the adsorbed and capillary waterare assumed separately, we follow the approach introduced by

Chung and Consolazio [10], where the diffusion of adsorbed water

is considered to be accounted for within the liquid water relativepermeability  K rw   (see  [13]). The main assumption in our model(46)–(56) is that the moisture transport in HSC at high temperaturerelated to the dry air pressure gradient is negligible when com-pared to the other causes, in particular, the vapour pressure andtemperature gradients, respectively (see [15]).

 A.3. Heat energy conservation equation

The local form of the macroscopic energy balance equation of the phase a  is [24]

qac a pDa

ha

Dt   þ r qac  ¼ Qa þ E a H aMa:   ðA:22Þ

Here, c a p is the specific isobaric heat,  qac  the heat flux due to conduc-

tion, Qa the volumetric heat source, E a represents energy exchangewith other phases and H a is the specific enthalpy of the phase  a.

Summing up the energy balances for all phases (a a; w; v ; s) of the multi-phase system, neglecting internal sources Qa and takinginto consideration

 PaE a ¼ 0 one arrives at the final form of the

Energy conservation equation for moist concrete as the multi- phase system:

ðqc  pÞ @ h

@ t  ¼ r qc  ðqc  pv Þ rh @ me

@ t   he @ md

@ t   hd;   ðA:23Þ

where

ðqc  pÞ ¼ c w p  qw gw þ c v 

 p  qv  g

v  þ c a p qa ga þ c s p qs gs;   ðA:24Þ

ðqc  pv Þ ¼ c w p qwgwv w þ c v 

 pqv g

v v v  þ c a pqagav a þ c s pqsgsv s;   ðA:25Þ

qc 

 ¼qs

c 

þqw

c 

 þqv 

c 

 þqa

c ;

  ðA:26

Þhe ¼ H v  H w;   ðA:27Þ

hd ¼ H w H s:   ðA:28ÞHere, c w p ; c v 

 p ; c a p  and c s p   [J kg1 K1] are the specific heats at constant

pressure of liquid water, water vapour, dry air and solid matrix,qw

c  ; qv 

c  ;qac   and   qs

c   represent heat fluxes due to conduction corre-sponding to liquid water, water vapour, dry air and solid matrix.Further, he   [J kg1] represents the enthalpy of evaporation per unitmass, while hd [J kg1] is the enthalpy of dehydration per unit mass.Finally, H w; H v  and H s [J kg1] are specific enthalpies of liquid water,water vapour and chemically bound water.

Neglecting the impact of diffusional flows of dry air and vapourin the gas mixture on heat transfer by convection yields

gwqwv w þ g g qv v v  ¼ q

v 

KK rg 

l g  |fflfflfflfflfflffl{zfflfflfflfflfflffl} v apour flow

  1 S BS w

qw

KK rw

lw

1 @ P c 

@ P v 

 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl} 

liquid water flow

S BS w

qwDB

@ S B@ P v  |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} 

adsorbed water diffusion

  M aM wP ahRðP v M w þ P aM aÞ Deff  |fflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflffl} 

v apour diffusion

0BBBB@

1CCCCArP v 

þ qv 

KK rg 

l g  |fflfflfflfflfflffl{zfflfflfflfflfflffl} v apour flow

  1 S BS w

qw

KK rw

lw |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl{zfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflffl} liquid water flow

þ   M aM wP v 

hRðP v M w þ P aM aÞ Deff  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflffl} v apour diffusion

0BBBB@

1CCCCArP a þ   1 S B

S w

qw

KK rw

lw

@ P c 

@ h |fflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflffl} liquid water flow

S BS w

qwDB

@ S B@ h |fflfflfflfflfflfflfflfflfflfflffl{zffl fflfflfflfflfflfflfflfflfflffl} 

adsorbed water diffusion

0BBB@

1CCCArh:   ðA:21Þ

M. Beneš, R. Štefan / International Journal of Heat and Mass Transfer 85 (2015) 110–134   133

7/23/2019 structural analysis2

http://slidepdf.com/reader/full/structural-analysis2 25/25

c v 

 pqv g

v v v  þ c a pqagav a ¼ ðc v 

 pqv g

v  þ c a pqagaÞv  g  ¼ c  g 

 pq g g g v  g ;   ðA:29Þwhere c  g 

 p  is the specific heat at constant pressure of gas mixture.

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